To estimate the sum of binomial coefficients, we note that Lemma 3 of [1]} yields the following. If \(n,m\in {\mathbb {N}}\)  are integers with \(2m<n+1\) , then
\( \binom{n}{m} \le \sum _{i=0}^m \binom{n}{i} \le \binom{n}{m}\frac{n-m+1}{n-2m+1}.\) 
